PharmPK Discussion - 3-compartmental model

On 10 May 2000 at 22:31:47, Chantal Le guellec (leguellec.aaa.med.univ-tours.fr) sent the message

Dear all,Could someone indicate me equations describing plasma concentration of adrug following 3-exponential pharmacokinetics and administered asIV-infusion, expressed as a function of intercompartmental constants andvolumes of each compartment.We were requested by anesthesiologists ofour institution who wish to implement such a model forcomputer-controlled infusion of anesthetic drugs.Many thanks for your help.

This is actually an answer from a previous discussion, butalso applies to your question:

Triexponential refers to the presence of three slopes in thelog plasma concentration versus time curve.

Cp = Ae-at + Be-bt + Ge-gt

In linear compartmental pharmacokinetics this is interpreted as athree-compartment model:

Xo \ k12 k13 Cpt2Cpt1Cpt3 k21 \ k31 \->k10

This is a linear mammillary model with elimination from a centralcompartment wich usually represents the plasma compartment. The systemis described by linear first order differential equations describingthe change in compartmental amounts of drug with time:

dX1/dt= -(k13+k10+k12)X1 + k21X2 +k31X3

dX2/dt= k12X1 - k21X2

dX3/dt= k13X1 -k31X3

This system of differential equations can be solved by Laplacetransforms and matrix algebra. The matix representation of the Laplacetransformed system of differential equations is:

[SI-A][Xs]= [Us]

Where the Laplace transform of the system of differential equationsabove is equal to the Laplace transformed vector of dose input, forexample: (Ko/s)(1-e-Ts).

This can be solved by matrix algebra to yield Laplace transformedquantities of each compartment, most important of which is the centralcompartment quantity (X1s):

X1s= [(Ko/s)(1-e-Ts)(s+k21)(s+k31)]/[(s+a)(s+b)(s+g)]

The inverse Laplace transform of the above results in an equationdescribing the amount in the central compartment as function of time,and dividing this by the Vc (volume of the central compartment) resultsin the equation for the plasma concentration over time:

Cp= Ko(k21-a)(k31-a)(1-e-aT)(e-at')/[a(b-a)(g-a)Vc] +

Ko(k21-b)(k31-b)(1-e-bT)(e-bt')/[b(a-b)(g-b)Vc] +

Ko(k21-g)(k31-g)(1-e-gT)(e-gt')/[g(a-g)(b-g)Vc]

The above equation describes the triexponential plasma concentrationcurve of a drug being administered by a an intermittent infusion, whereT= infusion time and t'= time after infusion.

Examples of drugs which exhibit triexpoential plasma concentrationcurves are vancomycin and aminoglycosides. However, this is difficult todetect as the curves really appear two compartment. Vancomycin has aninitial rapid distribution phase (T1/2~=7min) which can be detected withnumerous plasma samples and careful analysis. Aminglycosides have a longwashout phase due to renal tissue binding and release which can be detectedwith plasma samples taken long after the drug has been administered todetect a slow decline (T1/2~=100-200 hours) in plasma concentrations as drugis released from renal tissue.

Are you interested in fitting a sum of three exponential to the data andusing this to estimate plasma concentration, or fitting a three compartmentmodel to the data? If you are using sums of exponentials and fitting thisto concentration data, the equation is:

c(t) = A0 - A1*exp(-a1*t) - A2*exp(-a2*t) - A3*exp(-a3*t)

where A0 - A1 - A2 - A3 = 0 (assuming there is no drug in the system at timezero).

There are occasions, even when using a constant infusion directly intoplasma, wherethis constraint may have to be relaxed. In addition, there are occasionswhen a slightdelay is required. IN this case, t in the equation for c(t) is replaced byc - tlag where tlag is the time of the delay.

If you are using a three compartment model, you just need to structure yourmodel according to how you want the compartments hooked together, thensimulate the experiment on the model. Software such as SAAM II can doeither of these quite easily.

I recommend to use a model describing the behavior of the anestheticdrug during and after the end of a short- or long-time IV-infusion (1),instead of a model describing only a post infusion part of theconcentration-time profile of the drug. In general, such a model can be ofany order (lower or higher than a 3rd-order model). This model allows toutilize the information about the behavior of the drug contained in themeasurements over the whole time interval from the beginning of theinfusion to the end of the sampling and that even in the earliest phaserepresenting intravascular mixing. Moreover this model and a computer-controlled pump allow to adjust an adequate flow of the drug into thepatient body.1. Durisova M., et al., Meth. Find. Exp. Clin. Pharmacol., 1998, 20, 217.With best regards,Maria Durisova